Elements of Algebra

Abstract

In 1770, one of the founders of pure mathematics, Swiss mathematician Leonard Euler (1707–1783), published Elements of Algebra, a mathematics textbook for students. This edition of Euler's classic, published in 1822, is an English translation which includes notes added by Euler's tutor, Johann Bernoulli, and additions by Joseph-Louis Lagrange, both giants in eighteenth-century mathematics, as well as a short biography of Euler. Part 1 begins with elementary mathematics of determinate quantities and includes four sections on simple calculations (adding, subtracting, division, multiplication), and then progresses to compound calculations (fractions), ratios and proportions and algebraic equations. Part 2 consists of 15 chapters on analyses of indeterminate quantities. Here, Euler shows the reader several ways to solve polynomial equations up to the fourth degree. This landmark book showed students the beauty of mathematics, and more significantly, how to do it.

Unifying the Algebra for All Movement

“…Euler (1984) states, “Algebra has been defined, The science which teaches how to determine unknown quantities by means of those that are known ” (p. 186). While this simple definition succinctly summarizes the essence of algebra, the definition does not characterize the essential elements of algebra; that is, the key ideas necessary to establish the consistent topic rigor in courses and textbooks as argued by researchers (Cogan et al, 2001; Schmidt, 2002; Stein et al, 2011a).…”

Revisiting James Watt's Linkage with Implicit Functions and Modern Techniques

“…The equation (4) is known as the reduced cubic and it is the first step in the method of finding the general formula for the roots of the cubic, which [4] develops in full detail. We shall divide by z 3 and then define p = x and q = y.…”

“…Furthermore, for each point on this straight line the equation (4) holds. If we think of the plane as the (p, q) space of all cubic equations (4), then points on these straight lines are solutions of the equation (4). Hence, to solve a particular equation (4) we look to see which line(s), if any, pass through the point (p, q).…”

Chapter Two : Higher Variations of the Monty Hall Problem (3.0, 4.0) and Empirical Definition of the Phenomenon of Mathematics, in Boole��s Footsteps, as Something the Brain Does

“…How these numbers are obtained may be briefly reviewed below to make the present account fully self-sufficient. A particularly lucid and at the same time delightfully parsimonious presentation of the matter at hand is Euler's in his "Elements of Algebra" [9].…”

“…Because of this claim, influential writers of the early twentieth century such as Keynes dismissed Boole's work completely, and that hastened its progress into limbo. As Miller (2009) points out [9], however, Boole's solution was meaningful and complete, and based on adding an assumption that in absence of further information, and in particular, with no logical dependencies, an appropriate higher level of conditional statistical independence should be assumed. In modern day terms, Boole fitted judiciously chosen log linear models to the data, judiciously dropping higher order interactions about which there was no information anyway.…”